Quasi-Eulerian Hypergraphs

We generalize the notion of an Euler tour in a graph in the following way. An Euler family in a hypergraph is a family of closed walks that jointly traverse each edge of the hypergraph exactly once. An Euler tourthus corresponds to an Euler family with a single component. We provide necessary and sufficient conditions for the existence of an Euler family in an arbitrary hypergraph, and in particular, we show that every 3-uniform hypergraph without cut edges admits an Euler family. Finally, we show that the problem of existence of an Euler family is polynomial on the class of all hypergraphs.This work complements existing results on rank-1 universal cycles and 1-overlap cycles in triple systems, as well as recent results by Lonc and Naroski, who showed that the problem of existence of an Euler tour in a hypergraph is NP-complete.

On Tours that contain all Edges of a Hypergraph

Let $H$ be a $k$-uniform hypergraph, $k\geqslant 2$. By an Euler tour in $H$ we mean an alternating sequence $v_0,e_1,v_1,e_2,v_2,\ldots,v_{m-1},e_m,v_m=v_0$ of vertices and edges in $H$ such that each edge of $H$ appears in this sequence exactly once and $v_{i-1},v_i\in e_i$, $v_{i-1}\neq v_i$, for every $i=1,2,\ldots,m$. This is an obvious generalization of the graph theoretic concept of an Euler tour. A straightforward necessary condition for existence of an Euler tour in a $k$-uniform hypergraph is $|V_{odd}(H)|\leqslant (k-2)|E(H)|$, where $V_{odd}(H)$ is the set of vertices of odd degrees in $H$ and $E(H)$ is the set of edges in $H$. In this paper we show that this condition is also sufficient for hypergraphs of a broad class of $k$-uniform hypergraphs, that we call strongly connected hypergraphs. This result reduces to the Euler theorem on existence of Euler tours, when $k=2$, i.e. for graphs, and is quite simple to prove for $k>3$. Therefore, we concentrate on the most interesting case of $k=3$. In this case we further consider the problem of existence of an Euler tour in a certain class of $3$-uniform hypergraphs containing the class of strongly connected hypergraphs as a proper subclass. For hypergraphs in this class we give a sufficient condition for existence of an Euler tour and prove intractability (NP-completeness) of the problem in this class in general.

Topological Circles and Euler Tours in Locally Finite Graphs

We obtain three results concerning topological paths ands circles in the end compactification $|G|$ of a locally finite connected graph $G$. Confirming a conjecture of Diestel we show that through every edge set $E\in {\cal C}$ there is a topological Euler tour, a continuous map from the circle $S^1$ to the end compactification $|G|$ of $G$ that traverses every edge in $E$ exactly once and traverses no other edge. Second, we show that for every sequence $(\tau_i)_{i\in \Bbb N}$ of topological $x$–$y$ paths in $|G|$ there is a topological $x$–$y$ path in $|G|$ all of whose edges lie eventually in every member of some fixed subsequence of $(\tau_i)$. It is pointed out that this simple fact has several applications some of which reach out of the realm of $|G|$. Third, we show that every set of edges not containing a finite odd cut of $G$ extends to an element of $\cal C$.